We consider the problem of state-feedback stabilization for a class of sandwich systems, consisting of two linear systems connected in cascade via a saturation. In particular, we present design methodologies for constructing semiglobally and globally stabilizing controllers for such systems when the input is itself subject to saturation. The design is carried out under a set of assumptions that are proven to be both necessary and sufficient. The presented design methodologies are extensions of classical low-gain design methodologies developed for stabilizing linear systems subject to input saturation. The methodologies can be further extended to multilayer sandwich systems, consisting of an arbitrary number of cascaded linear systems with saturations sandwiched between them.